4. The Standard Model Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 4. The Standard Model 1 In this section... Standard Model particle content Klein-Gordon equation Antimatter Interaction via particle exchange Virtual particles Dr. Tina Potter 4. The Standard Model 2 The Standard Model Spin-1/2 fermions Charge (units of e) Quarks 2 u c t +3 d s b 1 −3 − − − Leptons e µ τ −1 νe νµ ντ 0 Plus antileptons and antiquarks Spin-1 bosons Mass ( GeV=c2) Gluon g 0 Strong force Photon γ 0 EM force W and Z bosons W ±; Z 91.2, 80.3 Weak force Spin-0 bosons Higgs h 125 Mass generation Dr. Tina Potter 4. The Standard Model 3 Theoretical Framework Macroscopic Microscopic Slow Classical Mechanics Quantum Mechanics Fast Special Relativity Quantum Field Theory The Standard Model is a collection of related Gauge Theories which are Quantum Field Theories that satisfy Local Gauge Invariance. Electromagnetism: Quantum Electrodynamics (QED) 1948 Feynman, Schwinger, Tomonaga (1965 Nobel Prize) Electromagnetism + Weak: Electroweak Unification 1968 Glashow, Weinberg, Salam (1979 Nobel Prize) Strong: Quantum Chromodynamics (QCD) 1974 Politzer, Wilczek, Gross (2004 Nobel Prize) Dr. Tina Potter 4. The Standard Model 4 The Schr¨odingerEquation To describe the fundamental interactions of particles we need a theory of Relativistic Quantum Mechanics p^2 Schr¨odingerEquation for a free particle E^ = 2m @ with energy and momentum operators E^ = i ; p^ = −irrr @t @ 1 (~ = 1 natural units) giving i = − rrr2 @t 2m which has plane wave solutions: (~r; t) = Ne−i(Et−p~:~r) 1st order in time derivative Not Lorentz Invariant! 2nd order in space derivatives Schr¨odingerequation cannot be used to describe the physics of relativistic particles. Dr. Tina Potter 4. The Standard Model 5 Klein-Gordon Equation Use the KG equation to describe the physics of relativistic particles. From Special Relativity: E 2 = p2 + m2 @ use energy and momentum operators E^ = i ; p^ = −irrr @t @2 @2 giving − = −rrr2 + m2 = (rrr2 − m2) Klein-Gordon @t2 @t2 Equation Second order in both space and time derivatives ) Lorentz invariant. Plane wave solutions (~r; t) = Ne−i(Et−p~:~r) but this time requiring E 2 = p~ 2 + m2, allowing E = ±pjp~j2 + m2 Negative energy solutions required to form complete set of eigenstates. ) Antimatter Dr. Tina Potter 4. The Standard Model 6 Antimatter and the Dirac Equation In the hope of avoiding negative energy solutions, Dirac sought a linear relativistic wave equation: @ i = (−i~α:rrr~ + βm) @t ~α and β are appropriate 4x4 matrices. is a column vector \spinor" of four wavefunctions. Two of the wavefunctions describe the states of a fermion, but the other two still have negative energy. Dirac suggested the vacuum had all negative energy states filled. A hole in the negative energy \sea" could be created by exciting an electron to a positive energy state. The hole would behave like a positive energy positive charged \positron". Subsequently detected. However, this only works for fermions... We now interpret negative energy states differently... Dr. Tina Potter 4. The Standard Model 7 Antimatter and the Feynman-St¨uckelberg Interpretation Consider the negative energy solution in which a negative energy particle travels backwards in time. e−iEt ≡ e−i(−E)(−t) Interpret as a positive energy antiparticle travelling forwards in time. Then all solutions can be used to describe physical states with positive energy, going forward in time. e.g. e+e− annihilation pair production e+ e− time time −−! γ γ −−! e− e+ All quantum numbers carried into a vertex by the e+ are the same as if it is regarded as an outgoing e−, or vice versa. Dr. Tina Potter 4. The Standard Model 8 Antimatter and the Feynman-St¨uckelberg Interpretation e− γ e− γ + − e− e =e γ − γ e− e time −−!time −−! The interpretation here is easy. The The emitted photon has more first photon emitted has less energy energy than the electron that than the electron it was emitted emitted it. Either view the top from. No need for \anti-particles" vertex as \emission of a negative or negative energy states. energy electron travelling backwards in time" or \absorption of a positive energy positron travelling forwards in time". Dr. Tina Potter 4. The Standard Model 9 Interaction via Particle Exchange Consider two particles, fixed at ~r1 and ~r2, which exchange a particle of mass m. e State i State j State i c a 1 p S pµ = (E;p ~) 2 E = Ej − Ei Time Calculate the shift in energy of state i due to this exchange (using second order perturbation theory): X hijHjjihjjHjii Sum over all possible states j ∆Ei = Ei − Ej j=6 i with different momenta where hjjHjii is the transition from i to j at ~r1 where hijHjji is the transition from j to i at ~r2 Dr. Tina Potter 4. The Standard Model 10 Interaction via Particle Exchange Consider hjjHjii (transition from i ! j by emission of m at ~r1) i = 1 2 Original 2 particles −i(Et−p~:~r) j = 1 2 3 3 = N e µ ∗ ∗ ∗ 3 represents a free particle with p = (E;p ~) normalise 1 1 = 2 2 = 3 3 = 1 p g= 2E is required on dimensional grounds, Let g be the probability of emitting m at r1 Z c.f. AQP vector potential of a photon. 3 ∗ ∗ ∗ g 3 hjjHjii = d ~r p 1 2 δ (~r − ~r1) Dirac δ function 1 2 3 Z 2E 3 3 g d ~rδ (~r − ~r1) i(Et−p~:~r1) = p Ne = 1 for ~r = ~r1 2E = 0 for ~r =6 ~r1 Similarly hijHjji is the transition from j to i at ~r2 g hijHjji = p Ne−i(Et−p~:~r2) 2E 2 2 ip~:(~r2−~r1) 2 2 ip~:(~r2−~r1) 1!2 X g N e X g N e Shift in energy state ∆Ei = = 2 2E Ei − Ej −2E (E = E −E ) j=6 i j=6 i j i Dr. Tina Potter 4. The Standard Model 11 Interaction via Particle Exchange Putting the pieces together Different states j have different momenta p~ for the exchanged particle. Therefore sum is actually an integral over all momenta: 3 Z g 2N2eip~:(~r2−~r1) Z g 2eip~:(~r2−~r1) 1 L ∆E 1!2 = ρ(p) dp = p2 dp dΩ i −2E 2 −2E 2 L3 2π r 3 3 ip~:(~r −~r ) 1 L 1 Z e 2 1 N = ; ρ(p) = p2 dΩ 2 2 L3 2π = −g 2 p dp dΩ 2π 2E E 2 = p2 + m2 The integral can be done by taking the z-axis along ~r = ~r2 − ~r1 Then p~:~r = pr cos θ and dΩ = 2π d(cos θ) 2 Z 1 2 ip~:~r −ip~:~r 1!2 g p e − e (see Appendix D) ∆Ei = − 2 2 2 dp 2(2π) 0 p + m ipr Write this integral as one half of the integral from −∞ to +1, which can be done by residues giving g 2 e−mr ∆E 1!2 = − i 8π r Dr. Tina Potter 4. The Standard Model 12 Interaction via Particle Exchange Final stage Can also exchange particle from 2 to 1: e State i State j State i c a 1 p S 2 Time g 2 e−mr Get the same result: ∆E 2!1 = − i 8π r Total shift in energy due to particle exchange is g 2 e−mr ∆E = − Yukawa Potential i 4π r Attractive force between two particles, decreasing exponentially with range r. Dr. Tina Potter 4. The Standard Model 13 Yukawa Potential g 2 e−mr V (r) = − Yukawa Potential 4π r Hideki Yukawa 1949 Nobel Prize Characteristic range = 1=m (Compton wavelength of exchanged particle) g 2 For m ! 0, V (r) = − infinite range (Coulomb-like) 4πr Yukawa potential with m = 139 MeV=c2 gives a good description of long range part of the interaction between two nucleons and was the basis for the prediction of the existence of the pion. Dr. Tina Potter 4. The Standard Model 14 Scattering from the Yukawa Potential Consider elastic scattering (no energy transfer) p⃗ f ⃗p Z p⃗i ip~:~r 3 Born Approximation Mfi = e V (r) d ~r qμ =( E ,⃗p) g 2 e−mr Yukawa Potential V (r) = − 2 2 2 4π r q =E −|⃗p| 2 2 Z −mr 2 q is invariant g e ip~:~r 3 g Mfi = − e d ~r = − “Virtual Mass” 4π r jp~j2 + m2 The integral can be done by choosing the z-axis along ~r, then p~:~r = pr cos θ and d3~r = 2πr 2 dr d(cos θ) For elastic scattering, qµ = (0;p ~); q2 = −|pj2 and exchanged massive particle is highly \virtual" g 2 M = fi q2 − m2 Dr. Tina Potter 4. The Standard Model 15 Virtual Particles Forces arise due to the exchange of unobservable virtual particles. The effective mass of the virtual particle, q2, is given by q2 = E 2 − jp~j2 and is not equal to the physical mass m, i.e. it is off-shell mass. The mass of a virtual particle can be +ve, -ve or imaginary. A virtual particle which is off-mass shell by amount ∆m can only exist for time and range 1 1 t ∼ ~ = ; range = ~ = = c = 1 ∆mc2 ∆m ∆mc ∆m ~ If q2 = m2, the the particle is real and can be observed. Dr. Tina Potter 4. The Standard Model 16 Virtual Particles For virtual particle exchange, expect a contribution to the matrix element of g 2 M = fi q2 − m2 where g Coupling constant g 2 Strength of interaction m2 Physical (on-shell) mass q2 Virtual (off-shell) mass 1 q2−m2 Propagator Qualitatively: the propagator is inversely proportional to how far the particle is off-shell.